Theorem dfsucon 39938

Description: 𝐴 is called a successor ordinal if it is not a limit ordinal and not the empty set. (Contributed by RP, 11-Nov-2023.)

Assertion

Ref	Expression
dfsucon	⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) ↔ ∃𝑥 ∈ On 𝐴 = suc 𝑥)

Distinct variable group:   𝑥,𝐴

Proof of Theorem dfsucon

Step	Hyp	Ref	Expression
1		3ancomb 1095	. . . 4 ⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴))
2		df-3an 1085	. . . 4 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ Lim 𝐴) ↔ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) ∧ ¬ Lim 𝐴))
3		df-ne 3017	. . . . . . . . 9 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
4	3	anbi2i 624	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) ↔ (Ord 𝐴 ∧ ¬ 𝐴 = ∅))
5	4	imbi1i 352	. . . . . . 7 ⊢ (((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ ((Ord 𝐴 ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥))
6		pm5.6 998	. . . . . . 7 ⊢ (((Ord 𝐴 ∧ ¬ 𝐴 = ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
7		iman 404	. . . . . . 7 ⊢ ((Ord 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) ↔ ¬ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
8	5, 6, 7	3bitrri 300	. . . . . 6 ⊢ (¬ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) ↔ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥))
9		dflim3 7562	. . . . . 6 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ¬ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
10	8, 9	xchnxbir 335	. . . . 5 ⊢ (¬ Lim 𝐴 ↔ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥))
11	10	anbi2i 624	. . . 4 ⊢ (((Ord 𝐴 ∧ 𝐴 ≠ ∅) ∧ ¬ Lim 𝐴) ↔ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) ∧ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
12	1, 2, 11	3bitri 299	. . 3 ⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) ↔ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) ∧ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
13		pm3.35 801	. . 3 ⊢ (((Ord 𝐴 ∧ 𝐴 ≠ ∅) ∧ ((Ord 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
14	12, 13	sylbi 219	. 2 ⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
15		eloni 6201	. . . . . 6 ⊢ (𝑥 ∈ On → Ord 𝑥)
16		ordsuc 7529	. . . . . 6 ⊢ (Ord 𝑥 ↔ Ord suc 𝑥)
17	15, 16	sylib 220	. . . . 5 ⊢ (𝑥 ∈ On → Ord suc 𝑥)
18		nlimsucg 7557	. . . . 5 ⊢ (𝑥 ∈ On → ¬ Lim suc 𝑥)
19		nsuceq0 6271	. . . . . 6 ⊢ suc 𝑥 ≠ ∅
20	19	a1i 11	. . . . 5 ⊢ (𝑥 ∈ On → suc 𝑥 ≠ ∅)
21	17, 18, 20	3jca 1124	. . . 4 ⊢ (𝑥 ∈ On → (Ord suc 𝑥 ∧ ¬ Lim suc 𝑥 ∧ suc 𝑥 ≠ ∅))
22		ordeq 6198	. . . . 5 ⊢ (𝐴 = suc 𝑥 → (Ord 𝐴 ↔ Ord suc 𝑥))
23		limeq 6203	. . . . . 6 ⊢ (𝐴 = suc 𝑥 → (Lim 𝐴 ↔ Lim suc 𝑥))
24	23	notbid 320	. . . . 5 ⊢ (𝐴 = suc 𝑥 → (¬ Lim 𝐴 ↔ ¬ Lim suc 𝑥))
25		neeq1 3078	. . . . 5 ⊢ (𝐴 = suc 𝑥 → (𝐴 ≠ ∅ ↔ suc 𝑥 ≠ ∅))
26	22, 24, 25	3anbi123d 1432	. . . 4 ⊢ (𝐴 = suc 𝑥 → ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) ↔ (Ord suc 𝑥 ∧ ¬ Lim suc 𝑥 ∧ suc 𝑥 ≠ ∅)))
27	21, 26	syl5ibrcom 249	. . 3 ⊢ (𝑥 ∈ On → (𝐴 = suc 𝑥 → (Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅)))
28	27	rexlimiv 3280	. 2 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅))
29	14, 28	impbii 211	1 ⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) ↔ ∃𝑥 ∈ On 𝐴 = suc 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∃wrex 3139  ∅c0 4291  Ord word 6190  Oncon0 6191  Lim wlim 6192  suc csuc 6193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-tr 5173  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197

This theorem is referenced by: (None)